Suppose we have measure spaces $(X_i, M_i, \mu_i), i=1, 2,3$, that are complete and $\sigma$-finite. I learned how to form a product measure from two measure spaces, but I wasn't so sure about product ...

I am learning about infinite (countable) product measure, which the exact statement of the theorem I write below. I was wondering if the theorem requires axiom of choice or not. I would appreciate any ...

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...

Hi all I was tackled by this question from Folland's real analysis second edition in the second chapter, it looks like a modified Egoroff theorem but I cannot really tackle it, it is question 41 of ...

I am learning about product measures and I was stuck on a detail of the proof. I would appreciate any assistance!
Suppose we have a measure spaces $(X_i, M_i, \mu_i), i=1, ..., N$, that are complete ...

Recall that if $\mathcal{A}\subset \mathcal{P}(X)$ is an algebra and $\mu_{0}:\mathcal{A} \to [0,\infty]$ is a premeasure on $\mathcal{A}$ then we can define the outer measure $\mu^{*}$ for any set ...

The part of the proof which I don't get is $$\nu(A)=\int_{A} g\ \mathsf d\mu$$ where $g$ is Radon-Nikodym derivative. He has a set of functions for which $$\int_{A} f\ \mathsf dx \le \nu(A) ,$$ he has ...

Is there a generalization of the Vitali-Hahn-Saks Theorem for nets of measures? I do not find any related literature. Take a sequence of bounded measures on a sigma-field and consider a subnet of this ...

Ex 1.1.9 in Tao's An introduction to measure theory asks us to show that any compact convex polytope in $\mathbb{R}^d$ is Jordan measurable. Is the following an efficient (or even valid) approach to ...

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B
Hi, in my summer real analysis (or measures and real analysis as my instructor refers ...

Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...

I was presented these two somewhat similar questions from Folland's real analysis (second edition) dealing with complex measures and their mutual singularity and absolute continuity. They are 3.19 and ...

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known.
Since this topology is too strong for my ...

Take the measure space to be $\mathbb{R}$ with Borel $\sigma$-algebra and Lebesgue measure (Although just thinking in terms of a general measure space probably works for this problem.)
Question: True ...

Answering this question it occurred to me that the OP's definition of integral is unsatisfactory in the following sense.
He defines it using the usual Lebesgue integral. I think it would be far more ...